5. (15 points) (Cobb-Douglas Utility) A consumer has the utility function of u($1, (2) = 1 1/4 3/4 The price of each good is denoted by pi and p2, and the consumer's income is denoted by m. (a) Mathematically express the consumer's constrained utility maximization problem and write down the corresponding Lagrangian function. (b) Derive the demand function for each good by using the Lagrangian optimization method. (c) Based on (b), calculate the price and income elasticities of good 2 (i.e., find 2:2 21 212 P2 and or2 m ) (d) Suppose now there is an ad-valorem tax t2 on good 2. Answer (a) and (b) again. (e) Based on (d), what is the optimal consumption bundle when p1 = 2, p2 = 2, t2 = 0.1 and m = 100? 6. (10 points) (Quasi-Linear Utility) A consumer has the utility function of u(1, 2) = In(x1) + X2. The price of each good is denoted by p, and p2, and the consumer's income is denoted by m. (a) Express x1 in terms of pi and p2. " (b) Derive the consumer's demand functions for goods 1 and 2. (c) Draw the income offer curve. (d) Draw Engel curves for goods 1 and 2 in different figures. (e) Based on (b), what is the optimal consumption bundle if the market prices and income are given as p1 = 1, p2 = 2, and m = 40? (f) Suppose the government imposes a value tax to on good 1. Answer (b) again with the tax. " 7Hint: Write an equation, which looks similar to: max y(X, z) s.t. Pad + PzZ = 100. 8Note: 1 is defined as in the LN. It is where the budget line is tangent to the indifference curve, [MRS| = P1/P2. "Hint: With the tax, 1 now satisfies |MRS| = (1 + 1)p1/p2. 3